Phase-Sensitive Cyclotron Frequency Measurements at Ultralow Energies

A novel technique for a direct and coherent measurement of the modified cyclotron frequency of an ion in a Penning trap at energies close to the thermal cooling limit is presented. This allows a rapid and both precise and accurate determination of the free-space cyclotron frequency in real Penning traps despite the existence of electric and magnetic field imperfections and relativistic shifts. The demonstrated performance paves the way for considerably improved bound-state $g$-factor measurements on the 10 ppt level and mass measurements in the 1 ppt range and possibly below.

Figure 1

Orthogonal and compensated cylindrical Penning trap with detection electronics and excitation ports. Two superconducting tank circuits allow for cooling of all eigenmodes. For details, see the text.

Figure 2

Excitation pulses scheme for the advanced integrated phase measurement method. For details, see the text.

Figure 3

Phase jitter due to finite thermal amplitudes at the beginning of the coupling process for a ${}^{28}\mathrm{Si}^{13+}$ ion cooled to thermal equilibrium with the axial detector. Coupling to the cyclotron detector would further reduce the systematic shifts by a factor of about 40. The red curve includes the limitation in the phase resolution arising from the SNR achievable in our setup due to electrostatic anharmonicities.

Figure 4

Unwrapping of the integrated phase allows a frequency calibration. The slope of the unwrapped phase represents the difference of the modified cyclotron frequency and the coupling frequency ${\nu }_{\mathrm{rf}}\simeq {\nu }_{+}+{\nu }_z$.

Figure 5

Fractional difference of consecutive measurements of the modified cyclotron frequency measured with the conventional mode coupling technique (a) and with the new phase-sensitive method (b). The fluctuation is reduced by more than an order of magnitude. The measurement time for (a) was 60 s, while (b) needs only 5 s. Further shielding of external fluctuations [19] and simultaneous coherent measurements will allow longer integration times and even better stability.